Why are two lines which are next to each other so different?

[#1]

The picture isn't too clear, but it looks to me like you've got a single blue/white pair coming into that box only.
A. That means the connection to the sky box is tapped off the BT line somewhere else - as knowdice says - and you need to find that location.
This has a knock-on implication - which is that it may be the other location that is the *real* master socket, and you NTE5 is really acting as a secondary (and possibly adding interference to boot)
Or it might be a badly-wired star arrangement.
It seems *highly* likely that this arrangement is going to be the source of your line's troubles. But sorting it out isn't necessarily straightforward - as legally, anything on the network side of the NTE5 (including the backplate) is solely BT's responsibility.
B. Where is the rest of the cable that the blue+white pair is part of? Is it a separate cable, and we can only see a bit of it?
For the life of me, it looks like it is going into the other box, and my suspicion would be that it is sharing the cable with your brother's line. That arrangement ought to mean that you're open for crosstalk problem between your two lines. That might limit your rates - but you'd see the same effect on both lines, so it isn't the sole cause of your troubles.
I guess a way to check is to see if one line improves when the other router is turned off. At that level of testing, you need to be tracking the detailed line statistics (Attenuation, SNR, sync rate, error counts) rather than just the sync rate, both before and after (It's probably worth posting those values here anyway).
I think you need to be tracing both lines further backwards, and the extension too. There is more to this story...

[#2]

I've exposed the other NTE5 and here's what's behind it:

Basically there is only one cable coming into both boxes - an orange pair going to my brother's socket, a blue pair going to my socket and a green pair not connected to anything.
I will try and work out where the extension box's cable comes from...
Here are the stats for my line at the moment:

-- DSL Information ---
DSL Driver Version:  3.4.4.10.0.1
DSL VPI/VCI:         0/38
DSL Status:          Up
DSL Mode:            G.dmt
DSL Channel:         0
DSL Upstream Rate:   448 kbps
DSL Downstream Rate: 2112 kbps
                      Down         up    
DSL Noise Margin:     190 dB        53 dB
DSL Attenuation:      315 dB       566 dB
DSL Transmit Power:  -901 dBm     -901 dBm

[#3]

Right - I've had a look outside and it appears to be what I assume is a star topography.
The little grey box mounted on the exterior wall where the lines come in from the telegraph pole has two black cables coming out of it.  One (thicker) cable goes to the two sockets in the picture in the posts above.  The other (slightly thinner) goes to the 'sky' extension socket.

[#4]

Quote from: Diggory

The other (slightly thinner) goes to the 'sky' extension socket.

That's very naughty.
Do you know who did it?
One "sort of" solution is to rejig via the spare pair, if you need the Sky socket working. Not elegant but it's possible.
Otherwise disconnect it inside the grey box.

[#5]

Thanks very much everyone for all your help.  Much appreciated.
I have disconnected the sky extension at the socket - I cannot get close to the BT little grey box yet as it's quite high up on my house.
So far the line appears quite stable - however it's been stable for extended periods before, so it's a bit early to tell if I have completely solved the issue.
Thanks again.